Large-scale real-space electronic structure calculations

Transcription

1 Large-scale real-space electronic structure calculations YIP: Quasi-continuum reduction of field theories: A route to seamlessly bridge quantum and atomistic length-scales with continuum Grant no: FA PI: Vikram Gavini Slides: 32 Vikram Gavini and Phani Motamarri Department of Mechanical Engineering University of Michigan, Ann Arbor

2 Defective Crystals Defects play a crucial role in influencing a variety of materials properties mechanical, electronic, optical, chemical Dislocations Vacancies/Interstitials Interfaces/Surfaces TEM image of dislocation partial T.J. Balk, K.J. Hemker, Phil. Mag. A, 2001 Prismatic loops formed from vacancies, Giess et. al, Microsc Microanal, 2005 TEM image of Ni-Al interface, Mann et.al, J. Appl. Phys Metal Plasticity Renders the strength of materials to 1/1000 its theoretical strength Creep, Spall, Ageing, hardening due to radiation Phase stability, Energetics, Diffusion mediation, defect sources and sinks

3 Defective crystals: The challenge The energetics of defects: (i) core-energy; (ii) elastic energy The core of a defect is governed by electronic structure need electronic structure calculations! Defects result in a vast span of interacting length scales Electronic structure of the core (10-12 m) Complex rearrangements of atoms around the core (10-9 m) Long ranged elastic effects (10-6 m) But need single physics at all length scales! - No patching, seamless description Realistic defect concentration in materials is parts per million! Challenge : Need electronic structure calculations at macroscopic scales! (i) Development of computational techniques for large-scale electronic structure calculation that can explicitly treat systems up to 10,000 atoms (ii) Development of seamless coarse-graining schemes using adaptive numerical schemes

4 Kohn-Sham density-functional theory (KSDFT) The KSDFT energy functional is given by, Local density approximation (LDA) Non-local Classical electrostatic interaction energy : Computed in Fourier-space (reciprocalspace) in almost all DFT implementations

5 KSDFT Real-space formulation (Suryanarayana & Gavini et al. JMPS 2010, ) Non-local Green s function for Laplace operator Electrostatic interactions can be re-written locally as, Thus, (Regularized nuclear charges)

6 KSDFT Real-space formulation (Suryanarayana & Gavini et al. JMPS 2010, ) The saddle-point problem is given by, Define, Theorem : Proof : Sobolev embeddings; Poincaré inequality (Direct Method)

7 Kohn-Sham eigenvalue problem Consider the E-L equation corresponding to the variational problem: Self consistent iteration (Kohn-Sham map) To avoid charge-sloshing:

8 State of the art Solutions to Kohn-Sham Equations Fourier Space Formulations Key Features (plane-waves) Very efficient for periodic calculations Restrictive to periodic domains Provide only uniform spatial resolution Suitable only when the solution fields are smooth. Real Space Formulations (LCAO, FDM, FEM) Key Features (LCAO) Suitable for isolated systems Can handle both pseudopotential and all electron calculations Systematic convergence can not be ascertained Parallel scalability is a concern

9 KSDFT FE discretization Use finite-element basis for computing u 1 i=1 i=2 N 1 (r) N 2 (r) N 3 (r) i=1 i=2 r r Features of finite-element basis: 1. Unstructured coarse-graining 2. Complex geometries can be represented, and arbitrary boundary conditions can be imposed. 3. Systematic convergence 4. Ease of parallel implementation By changing the positioning of the nodes the spatial resolution of basis can be changed/adapted

10 KSDFT FE discretization Main Limitations: Previous attempts showed that the number of FE basis functions (linear) needed to obtain chemical accuracy is very large ~ 100,000-1,000,000 basis functions per atom. The finite-element discretization leads to a generalized eigenvalue problem, which is more challenging to solve than a standard eigenvalue problem Present Work: We demonstrate an efficient, scalable computational approach using adaptive higher-order finite-element discretization. We propose a linear scaling algorithm (in number of electrons) which treats both insulating and metallic systems on an equal footing.

11 KSDFT FE discretization Discrete eigenvalue problem: Transformation to a standard eigenvalue problem: Remark: denotes the projection of the Hamiltonian operator into a space spanned by Löwden orthonormalized finite-element basis

12 Can higher-order finite-elements do any better? Here, we investigate the viability and computational efficiency afforded by higher-order finite-element discretization in electronic structure calculations using density functional theory to answer the following questions: What is the numerical convergence rate for various orders of finite-element approximations in electronic structure calculations using DFT? What is the computational advantage derived by using higher-order finite element discretization in terms of the CPU time? We present some of the first studies which demonstrate the computational efficiency afforded by higher-order elements for Kohn-Sham DFT calculations.

13 Rate of convergence of the finite element approximation The study has been carried out by a suite of higher order elements: TET 10 (TETRAHEDRAL QUADRATIC ELEMENT) HEX 27 (TRI QUADRATIC HEXAHEDRAL ELEMENT) HEX 64 (TRI CUBIC HEXAHEDRAL ELEMENT) HEX 125 (TRI QUARTIC HEXAHEDRAL ELEMENT) HEX 64 SPECTRAL, HEX 125 SPECTRAL upto 10 th order (Lagrange Polynomials are constructed on Gauss-Lobatto Legendre Points for spectral elements) Elements have been tested against three types of problems: (a) CH 4 (b) Barium Cluster (35 atoms) (a) CH 4 : An all electron calculation (b) Barium Cluster: Pseudopotential calculation

14 Convergence rates (Motamarri et al. J. Comp. Phys (2013)) Barium Cluster Methane Molecule Optimal rate of convergence!

15 Computational efficiency of higher-order FE discretization Two Key questions: How do higher-order FE discretizations compare to lower order elements in computational efficiency? How do higher-order FE discretizations compare against plane-wave basis and Gaussian basis? Key ideas in improving computational efficiency: Developed a priori mesh adaption techniques Use of Gauss-Legendre-Lobatto quadrature rules for the overlap matrix in conjunction with Spectral FE discretization Developed a Chebyschev acceleration technique to directly compute the eigenspace

16 A priori mesh adaption (Motamarri et al. J. Comp. Phys (2013)) Error Estimate: Optimal mesh distribution:

17 Spectral FE and Gauss-Lobatto-Legendre quadrature Spectral-element basis functions: Constructed from Lagrange polynomials interpolated through nodes corresponding to the roots of the derivatives of the Legendre polynomials and boundary nodes (GLL points) Upon using a Gauss-Lobatto-Legendre quadrature rule, the quadrature points coincide with the FE nodes Remarks: Transformation to standard eigenvalue problem is trivial The reduced order quadrature rule is only employed for the computation of the overlap matrix, and the full Gauss quadrature is employed to compute the Hamiltonian matrix.

18 Eigen-space computation: Chebyschev acceleration (Motamarri et al. J. Comp. Phys (2013)) E Occupied eigen-space Unwanted eigen-space Occupied eigen-space -1 1

19 Numerical algorithm 1. Start with initial guess for electron density and the initial wavefunctions 2. Compute the discrete Hamiltonian using the input electron density 3. Compute the Chebyshev filtered basis : 4. Orthonormalize the basis and compute, the projected Hamiltonian into the subspace spanned by 5. Compute the Fermi-energy and the output electron density by diagonalizing and using the following equation 6. If, EXIT; else, compute new using a mixing scheme and go to (2).

20 Computational efficiency (Motamarri et al. J. Comp. Phys. 2013) Barium Cluster Methane Molecule

21 Aluminum clusters

22 All-electron calculations 100 atom Graphene sheet

23 Scalability

24 Computational complexity Complexity in each SCF iteration: M: Number of degrees of freedom N: Number of electrons (M N) Chebyshev filtering procedure : O(MN) Orthonormalization of Chebyschev filtered vectors : O(MN 2 ) Diagonalization of the projected Hamiltonian: O(N 3 ) Cubic Scaling in N!

25 Subspace projection technique (Motamarri & Gavini, Phys. Rev. B (2014)) Key features of the proposed method: Chebyshev filtering to generate the approximate occupied subspace Construct non-orthogonal localized basis functions that span the same space & truncate these localized functions beyond a prescribed tolerance Subsequently localized basis functions have a compact support Use an adaptive tolerance for the truncation: truncation tolerance tied to the error in the SCF iteration; ensures strict control on accuracy Project Hamiltonian into the occupied subspace expressed in the nonorthogonal basis Fermi-operator expansion of the projected Hamiltonian to estimate Fermienergy and compute the electron density Avoids diagonalization of the Hamiltonian to compute orbital occupancies Applicable for both metallic and insulating systems Applicable for both pseudopotenial and all-electron calculations

26 Subspace projection approach: Key ideas Eigen-space from Chebyschev filtering Localized basis spanning eigen-space (Garcia-Cervera et al.) Project Hamiltonian in the localized basis: Remarks: Locality of is sparse and can be computed in O(N) complexity can be computed using Newton-Schultz algorithm which has O(N) complexity Finally, can be computed in O(N), if is sparse.

27 Subspace projection approach: Key ideas Computation of electron density Recall: No diagonalization No knowledge of eigenvalues and eigenvectors Compute density matrix instead: The electron density is the diagonal of the density matrix Fermi-operator expansion techniques can be employed to compute the density matrix: Challenge: ; spectral width of the discrete Hamiltonian is about !

28 Subspace projection approach: Key ideas Fermi-operator expansion of the projected Hamiltonian: Compute the density matrix using the projected Hamiltonian in the nonorthogonal localized basis The spectral width of the projected Hamiltonian is ~ O(10) and thus can efficiently employ the Fermi-operator expansion This approach treats both insulating and metallic systems on equal footing This approach is applicable for both pseudopotential calculations and allelectron calculations

29 Case study: Al nano-clusters (3x3x3 9x9x9) Pseudopotential calculations Total computational time Subspace projection scaling: O(N 1.46 ) Electron density contours on the midplane of the 9x9x9 nano-cluster Accuracy of subspace projection method commensurate with chemical accuracy

30 Case study: Alkane chains (C 33 H 68 C 2350 H 4702 ) Pseudopotential calculations Numerical accuracy Total computational time Isocontours of alkane chain C 900 H 1802 Subspace projection scaling: O(N 1.18 )

31 Case study: Si nano-clusters (1x1x1 3x3x3) All-electron calculations Total computational time Subspace projection scaling: O(N 1.85 ) Electron density contours on the midplane of the 3x3x3 Si nano-cluster Accuracy of subspace projection method commensurate with chemical accuracy

32 Ongoing/future work Real-space DFT-FE: Develop robust geometry optimization techniques Incorporate more advanced exchange-correlation functionals (beyond LDA, GGA) Exploring tensor structured techniques and low rank approximations in conjunction with real-space formulation Coarse-graining KSDFT: Localization of the wavefunctions is key for extending the coarse-graining ideas O(N) formulations: Non-orthogonal localized orbitals QC-KSDFT: localization -> predictor-corrector approach -> QC Electronic structure calculations at macroscopic scales with Kohn-Sham DFT will enable a quantum-mechanically accurate study of defects in materials

33 Concluding remarks Developed real-space formulation for Kohn-Sham DFT Reformulation of electrostatics as a local variational problem Mathematical analysis Finite-element discretization of Kohn-Sham DFT & Numerical algorithms Optimal rates of convergence Spectral elements in conjunction with GLL quadratures (for overlap matrix) Chebyshev filtering to directly compute the eigenspace Large-scale calculations possible Algorithms exhibit good scalability Development of a linear-scaling algorithm Localized basis spanning the Chebyshev filtered subspace Project of Hamiltonian into subspace Use Fermi-operator expansion on the projected Hamiltonian

34 Publications from AFOSR Comp Math program support P. Motamarri, T. Blesgen, V. Gavini, Tucker-tensor algorithm for large-scale Kohn-Sham density functional theory calculations, submitted (2015). P. Motamarri, V. Gavini, Phys. Rev. B 90, (2014). M. Iyer, T.M. Pollock, V. Gavini, Phys. Rev. B 89, (2014). P. Motamarri, M.R. Nowak, K. Leiter, J. Knap, V. Gavini, J. Comput. Phys. 253, (2013). (Previous award) T. Blesgen, V. Gavini, V. Khoromskaia, J. Comput. Phys. 231, (2012). P. Mottamari, M. Iyer, J. Knap, V. Gavini, J. Comput. Phys. 231, (2012). V. Gavini, L. Liu, J. Mech. Phys. Solids 59, (2011). M. Iyer, V. Gavini, J. Mech. Phys. Solids 59, (2011). P. Suryanarayana, P., V. Gavini, T. Blesgen, K. Bhattacharya, M. Ortiz, J. Mech. Phys. Solids 58, (2010). B. Radhakrishnan, V. Gavini, Phys. Rev. B 82, (2010). V. Gavini, Proc. Roy. Soc. A. 465, (2009).

35 THANK YOU!